Let A denote Toeplitz algebra. K(H²) is liminal. [ A/( K(H²))] is *-isomorphic to C(T), so it is abelian and therefore liminal. Hence A is postliminal .But identity representation of A on H² is irreducible and not finite dimensional, so A isn't liminal. For details see [Mur, Example 5.6.4].

Ref.
[Mur] G.J. Murphy, C^*-algebras and operator theory, Academic Press, 1990.